Methods and Apparatus for Deriving an Order-16 Integer Transform

ABSTRACT

Apparatus, systems and techniques based on an integer transform for encoding and decoding video or image signals, including an order-16 integer transform from a Microsoft Media Video order-8 integer transform with a high energy-packing ability and an improved data compression in the field of image and video coding. For example, a method and an apparatus are provided for deriving an order-16 integer transform from an order-8 integer transform in the standard transform of Microsoft Media Video. Eight additions and eight subtractions are used to assign the data elements to be transformed to an intermediate matrix; and then two fast algorithms for the computation of the order-8 transform may be applied to the first 8 vectors of the intermediate matrix, and the last 8 vectors of the intermediate matrix, respectively. The derived order-16 integer transform tends to produce small magnitude and high frequency transformed coefficients, and thus achieve high compressibility.

BACKGROUND

This application relates to digital image and video processing.

Various applications exist for digital video communication and storage,and corresponding international standards have been and are continuingto be developed. To achieve low bit rate communications, MPEG-2, MPEG-4Part 2, and H.263 standards divide a picture into 16×16 non-overlappingblocks called macro block and then use 2 dimensional (2D) order-8discrete cosine transform (DCT) in digital video compression algorithmsused in the coding standards. Recently, new video coding standards, suchas H.264/AVC, AVS and SMPTE 421M etc, have been proposed that use 2Dorder-4 or order-8 transforms to provide a better compressibility thanthe video standards based on the 2D order-8 DCT. In those standards, the2D order-4 or order-8 transforms are used so as to seek a trade-offbetween computational efficiency and coding efficiency.

SUMMARY

This application describes examples and implementations of apparatus,systems and techniques based on an integer transform for encoding anddecoding video or image signals, including an order-16 integer transformfrom a Microsoft Media Video order-8 integer transform with a highenergy-packing ability and an improved data compression in the field ofimage and video coding.

In one aspect, a process for deriving an order-16 integer transform froman order-8 integer transform in image and video coding includes:

retrieving a data matrix X_(16×16) to be transformed;

assigning the retrieved matrix X_(16×16) to an intermediate matrixC_(16×16)by eight additions and eight subtractions; and

deriving a resultant matrix Z_(16×16) by carrying out the order-8integer transform twice by a rule of

[z _(0,m) z _(1,m) . . . z _(7,m)]^(T) =E _(8×8) [c _(0,m) c _(1,m) . .. c _(7,m)]^(T)

[z _(8,m) z _(9,m) . . . z _(15,m)]^(T) =E _(8×8) [c _(8,m) c _(9,m) . .. c _(15,m)]^(T)

wherein z_(n,m) is of the (n,m)th element of the matrix Z_(16×16), andE_(8×8) is a standard order-8 transform used in the standard ofMicrosoft Media Video.

In another aspect, an apparatus for deriving an order-16 integertransform from an order-8 integer transform in image and video codingincludes:

an assignment unit configured to retrieve a data matrix X_(16×16) to betransformed, and then to assign the retrieved matrix X_(16×16) to anintermediate matrix C_(16×16) by eight additions and eight subtractions;and

a transform unit configured to derive a resultant matrix Z_(16×16) bycarrying out the order-8 integer transform twice by a rule of

[z _(0,m) z _(1,m) . . . z _(7,m)]^(T) =E _(8×8) [c _(0,m) c _(1,m) . .. c _(7,m)]^(T)

[z _(8,m) z _(9,m) . . . z _(15,m)]^(T) =E _(8×8) [c _(8,m) c _(9,m) . .. c _(15,m)]^(T)

wherein z_(n,m) is of the (n,m)th element of the matrix Z_(16×16), andE_(8×8) is a standard order-8 transform used in the standard ofMicrosoft Media Video.

In some implementations, only additions and subtraction operations areperformed in the processing. Hence, if an order-8 transform is aninteger transform, the resultant order-16 integer transform is also aninteger transform and thus can be easily implemented with good precisionif the integers are small. In addition, the derived order-16 transformbased on what is disclosed here can pack more energy into low frequencycoefficients, i.e. the transformed coefficient z_(u,v) with small u andv, and leave little energy to high frequency coefficients, i.e. thetransformed coefficient z_(u,v) with large u and v in some cases. Hence,it is an effective tool to improve the compression ability of a coder.

The details of the above and other aspects of the described apparatus,systems and techniques are set forth in the accompanying drawings, thedescription and claims below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example of a flow chart of a process for derivinga forward 2D order-16 integer transform from an order-8 integertransform;

FIG. 2 illustrates an example of a fast algorithm of 1D order-16 integertransform carried out at the step 102 of FIG. 1;

FIG. 3 illustrates an example of an alternative fast algorithm of 1Dorder-16 integer transform carried out in the process shown in FIG. 1;and

FIG. 4 illustrates an example of an apparatus for deriving a forward 2Dorder-16 integer transform from an order-8 integer transform.

DETAILED DESCRIPTION

An exemplary a process for deriving an order-16 transform from anorder-8 transform is now described.

FIG. 1 illustrates a process 100 for deriving a forward 2D order-16integer transform from an order-8 transform. The process 100 begins atstep 101, where a data matrix X_(16×16) is retrieved from video or imagesignal stream. As an example, the data matrix X_(16×16) is given asbelow

$\begin{matrix}{X_{16 \times 16} = {\begin{bmatrix}x_{0,0} & x_{0,1} & \cdots & x_{0,15} \\x_{1,0} & x_{1,1} & \; & x_{1,15} \\\vdots & \; & ⋰ & \vdots \\x_{15,0} & x_{15,1} & \cdots & x_{15,15}\end{bmatrix}.}} & (1)\end{matrix}$

At step 102, a 1 D (one dimension) integer transform is carried out foreach of the 16 column vectors. For the purpose of description, let X_(m)be one vector of the 16 column vectors of the matrix X_(16×16), which isdenoted as

X _(m) =[x _(0,m) x _(1,m) . . . x _(15,m)]^(T)   (2)

Herein, the resultant of 1D integer transform is denoted as a matrixZ_(16×16). Let vector Z_(m) be one vector of the matrix Z_(16×16),vector Z_(m)=E_(16×16)X_(m), i.e.

$\begin{matrix}{Z_{m} = {\begin{bmatrix}z_{0,m} \\z_{1,m} \\\vdots \\z_{15,m}\end{bmatrix} = {{E_{16 \times 16}X_{m}} = {E_{16 \times 16}\begin{bmatrix}x_{0,m} \\x_{1,m} \\\vdots \\x_{15,m}\end{bmatrix}}}}} & (3)\end{matrix}$

wherein E_(16×16) is an order-16 transform, which is to be explained indetail later.

To improve the computational efficiency, a fast algorithm can be used toderive an order-16 integer transform from a conventional order-8transform. In one implementation, the fast algorithm can include:

a) eight additions and eight subtractions; and

b) computing the order-8 transform E_(8×8) twice by using a known fastalgorithm of the order-8 transform.

Specific examples of the fast algorithm are discussed below.

EXAMPLE 1

This Example will be discussed in referring to FIG. 2, in which anexample of a fast algorithm of 1D order-16 integer transform carried outat step 102 of FIG. 1 is illustrated.

Referring to FIG. 2, the vector X_(m) is firstly transformed into anintermediate vector C_(m) by eight additions and eight subtractions asgiven in equation (4),

$\begin{matrix}{C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\cdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\\cdots \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\cdots \\{x_{14,m} + x_{15,m}} \\{x_{0,m} - x_{1,m}} \\{x_{2,m} - x_{3,m}} \\\cdots \\{x_{14,m} - x_{15,m}}\end{bmatrix}}} & (4)\end{matrix}$

where c_(j,m) is the (j)th element of the vector C_(m), and x_(j,m) isthe O)th element of the vector X_(m), for 0≦j≦15.

That is, c_(i,m)=x_(2i,m)+X_(2i+1,m), C_(i+8,m)=X_(2i,m)−X_(2i+1, m)where 0≦i≦7.

Then, a resultant matrix Z_(16×16) is calculated by following twoequations:

[z _(0,m) z _(1,m) . . . z _(7,m)]^(T) =E _(8×8) [c _(0,m) z _(1,m) . .. c _(7,m)]^(T)   (5.1)and

[z _(15,m) z _(14,m) . . . z _(8,m)]^(T) =E _(8×8) [c _(8,m) c _(9,m) .. . c _(15,m)]^(T)   (5.2)

where z_(j,m) is the (j,m)th element of the matrix Z_(16×16) (0≦m≦15),and E_(8×8) is a standard transform used in Windows Media Video, whichis shown as

$\begin{matrix}{E_{8} = {\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 15 & 9 & 4 & {- 4} & {- 9} & {- 15} & {- 16} \\16 & 6 & {- 6} & {- 16} & {- 16} & {- 6} & 6 & 16 \\15 & {- 4} & {- 16} & {- 9} & 9 & 16 & 4 & {- 15} \\12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 \\9 & {- 16} & 4 & 15 & {- 15} & {- 4} & 16 & {- 9} \\6 & {- 16} & 16 & {- 6} & {- 6} & 16 & {- 16} & 6 \\4 & {- 9} & 15 & {- 16} & 16 & 15 & 9 & {- 4}\end{bmatrix}.}} & (6)\end{matrix}$

Since the order-8 transform is well known to those skilled in the art,the detailed discussion thereof is omitted for the sake of space.

The transformation of the data matrix X_(16×16) into the matrixZ_(16×16) using the above order-8 transform can be represented as oneorder-16 transform by the following equation:

$\begin{matrix}{{Z_{16 \times 16} = {E_{16 \times 16}X_{16 \times 16}}}{{where},}} & (7) \\{{E_{16 \times 16} = {\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 16 & 15 & 15 & 9 & 9 & 4 & 4 & {- 4} & {- 4} & {- 9} & {- 9} & {- 15} & {- 15} & {- 16} & {- 16} \\16 & 16 & 6 & 6 & {- 6} & {- 6} & {- 16} & {- 16} & {- 16} & {- 16} & {- 6} & {- 6} & 6 & 6 & 16 & 16 \\15 & 15 & {- 4} & {- 4} & {- 16} & {- 16} & {- 9} & {- 9} & 9 & 9 & 16 & 16 & 4 & 4 & {- 15} & {- 15} \\12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 & 12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 \\9 & 9 & {- 16} & {- 16} & 4 & 4 & 15 & 15 & {- 15} & {- 15} & {- 4} & {- 4} & 16 & 16 & {- 9} & {- 9} \\6 & 6 & {- 16} & {- 16} & 16 & 16 & {- 6} & {- 6} & {- 6} & {- 6} & 16 & 16 & {- 16} & {- 16} & 6 & 6 \\4 & 4 & {- 9} & {- 9} & 15 & 15 & {- 16} & {- 16} & 16 & 16 & {- 15} & {- 15} & 9 & 9 & {- 4} & {- 4} \\4 & {- 4} & {- 9} & 9 & 15 & {- 15} & {- 16} & 16 & 16 & {- 16} & {- 15} & 15 & 9 & {- 9} & {- 4} & 4 \\6 & {- 6} & {- 16} & 16 & 16 & {- 16} & {- 6} & 6 & {- 6} & 6 & 16 & {- 16} & {- 16} & 16 & 6 & {- 6} \\9 & {- 9} & {- 16} & 16 & 4 & {- 4} & 15 & {- 15} & {- 15} & 15 & {- 4} & 4 & 16 & {- 16} & {- 9} & 9 \\12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} & 12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} \\15 & {- 15} & {- 4} & 4 & {- 16} & 16 & {- 9} & 9 & 9 & {- 9} & 16 & {- 16} & 4 & {- 4} & {- 15} & 15 \\16 & {- 16} & 6 & {- 6} & {- 6} & 6 & {- 16} & 16 & {- 16} & 16 & {- 6} & 6 & 6 & {- 6} & 16 & {- 16} \\16 & {- 16} & 15 & {- 15} & 9 & {- 9} & 4 & {- 4} & {- 4} & 4 & {- 9} & 9 & {- 15} & 15 & {- 16} & 16 \\12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12}\end{bmatrix}.}}\;} & (8)\end{matrix}$

EXAMPLE 2

This Example will be discussed in referring to FIG. 3, which illustratesanother example of a fast algorithm of 1D order-16 integer transformcarried out at step 102 of FIG. 1.

Referring to FIG. 3, the vector X_(m) is firstly transformed into anintermediate vector C_(m) by eight additions and eight subtractions asgiven in equation (9),

$\begin{matrix}{C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\vdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\c_{10,m} \\c_{11,m} \\c_{12,m} \\c_{13,m} \\c_{14,m} \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\vdots \\{x_{14,m} + x_{15,m}} \\{x_{0,m} - x_{1,m}} \\{x_{3,m} - x_{2,m}} \\{x_{4,m} - x_{5,m}} \\{x_{7,m} - x_{6,m}} \\{x_{8,m} - x_{9,m}} \\{x_{11,m} - x_{10,m}} \\{x_{12,m} - x_{13,m}} \\{x_{15,m} - x_{14,m}}\end{bmatrix}}} & (9)\end{matrix}$

where c_(j,m) is the (j)th element of the vector C_(m), and x_(j,m) isthe O)th element of the vector X_(m), for 0≦j≦15.

Then, a resultant matrix Z_(16×16) is calculated by using the standardtransform E_(8×8) used in Windows Media Video, according to the abovementioned equations (5.1) and (5.2).

Then, the transformation of the data matrix X_(16×16) into the matrixZ_(16×16) using the above order-8 transform E_(8×8) can be representedas one order-16 transform by the following equation:

$\begin{matrix}{{Z_{16 \times 16} = {E_{16 \times 16}X_{16 \times 16}}}{{where},}} & (10) \\{E_{16 \times 16} = {\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 16 & 15 & 15 & 9 & 9 & 4 & 4 & {- 4} & {- 4} & {- 9} & {- 9} & {- 15} & {- 15} & {- 16} & {- 16} \\16 & 16 & 6 & 6 & {- 6} & {- 6} & {- 16} & {- 16} & {- 16} & {- 16} & {- 6} & {- 6} & 6 & 6 & 16 & 16 \\15 & 15 & {- 4} & {- 4} & {- 16} & {- 16} & {- 9} & {- 9} & 9 & 9 & 16 & 16 & 4 & 4 & {- 15} & {- 15} \\12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 & 12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 \\9 & 9 & {- 16} & {- 16} & 4 & 4 & 15 & 15 & {- 15} & {- 15} & {- 4} & {- 4} & 16 & 16 & {- 9} & {- 9} \\6 & 6 & {- 16} & {- 16} & 16 & 16 & {- 6} & {- 6} & {- 6} & {- 6} & 16 & 16 & {- 16} & {- 16} & 6 & 6 \\4 & 4 & {- 9} & {- 9} & 15 & 15 & {- 16} & {- 16} & 16 & 16 & {- 15} & {- 15} & 9 & 9 & {- 4} & {- 4} \\12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 \\16 & {- 16} & {- 15} & 15 & 9 & {- 9} & {- 4} & 4 & {- 4} & 4 & 9 & {- 9} & {- 15} & 15 & 16 & {- 16} \\16 & {- 16} & {- 6} & 6 & 6 & {- 6} & 16 & {- 16} & {- 16} & 16 & {- 6} & 6 & 6 & {- 6} & {- 16} & 16 \\15 & {- 15} & {- 4} & 4 & {- 16} & 16 & 9 & {- 9} & 9 & {- 9} & {- 16} & 16 & {- 4} & 4 & 15 & {- 15} \\12 & {- 12} & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & {- 12} & 12 \\9 & {- 9} & 16 & {- 16} & {- 4} & 4 & {- 15} & 15 & {- 15} & 15 & {- 4} & 4 & 16 & {- 16} & 6 & {- 6} \\6 & {- 6} & 16 & {- 16} & 16 & {- 16} & 6 & {- 6} & {- 6} & 6 & {- 16} & 16 & {- 16} & 16 & {- 6} & 6 \\4 & {- 4} & 9 & {- 9} & 15 & {- 15} & 16 & {- 16} & 16 & {- 16} & 15 & {- 15} & 9 & {- 9} & 4 & {- 4}\end{bmatrix}.}} & (11)\end{matrix}$

A 2D fast algorithm for the order-16 integer transform can be achievedby performing the 1D integer transform in vertical for each column inX_(16×16) to form matrix Z_(16×16) of elements z_(i,j) and then inhorizontal for each row in Z_(16×16) to form matrix Z_(16×16)′ ofelements Z_(i,j)′ as given in equation (12)

Z _(16×16) E _(16×16) X _(16×16) E _(16×16)   (12).

Specifically, referring to FIG. 1 again, at step 102, for each of 16column vectors of the matrix X_(16×16), the above mentioned 1D integertransform is carried out to achieve an intermediate matrix Z_(16×16),and then at step 103, the above mentioned 1D integer transform iscarried out again for each of the 16 row vectors of the intermediatematrix Z_(16×16) so as to achieve the resultant matrix Z_(16×16)′.

In FIG. 1, step 102 is performed before step 103. Alternatively, one maychoose to perform step 103 before step 102. In other words, one mayperform the 1D transform in horizontal for each row in X_(16×16) first.Such step is represented by the operation X_(16×16) E^(T) _(16×16) in(12). It is then followed by the 1D transform in vertical, which isrepresented by the multiplication of E_(16×16) in (12). The sameZ_(16×16)′ can be obtained in either way.

As is well known in the art, a transform is good if it can pack moreenergy into low frequency coefficients (i.e. z′_(u,v) with small u, v)and leave little energy to high frequency coefficients (i.e. z′_(u,v)with large u, v). In general, a transform with its low frequency basisvectors (i.e. the first few row of E_(16×16)) resembling a slowlychanging vector x_(i) of the matrix X_(16×16) is good. As is shown inequation (8), the first three basis vectors of transform E_(16×16) areas follows

E ₁==[16 16 15 15 9 9 4 4 −4 −4 −9 −9 −15 −15 −16 −16]E ₂=[16 16 6 6 −6−6 −16 −16 −16 −16 −6 −6 6 6 16 16]E ₃=[15 15 −4 −4 −16 −16 −9 −9 9 9 1616 4 4 −15 −15].

In most cases, most of the energy of X_(16×16) is packed into lowfrequency coefficients related to the first several basis vectors oftransform E_(16×16) like E₁, E₂, and E₃ etc. The remaining small amountof energy of X_(16×16) is represented by other coefficients which becomezero after quantization. As a result, higher compression rates can beachieved.

Then, the process 100 goes to step 104, where a quantization is carriedout to convert the transformed matrix Z′_(16×16) into a matrix ofquantized transform Y_(Q)(i,j), which requires less bits forrepresentation. The amount of the bit reduction is controlled by aquantization parameter denoted as QP, which is shown as follows:

Then, at step 105, the quantized matrix Y_(Q)(i, j) is scaled by ascaling matrix K_(16×16) to form a matrix Y_(16×16), as given inequation (13)

y _(i,j) =Y _(Q)(i,j)·K _(16×16) ² (i,j)   (13)

where y_(i,j) is the (i,j)th element of matrix Y_(16×16), and K_(16×16)(i,j) is the (i,j)th element of K_(16×16). K_(16×16) is derived from ascaling matrix K_(8×8), which is listed as follows:

$\begin{matrix}{K_{8 \times 8} = {\begin{bmatrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{bmatrix}\mspace{14mu} {where}\mspace{14mu} \left\{ {\begin{matrix}{a = \frac{1}{\sqrt{1152}}} \\{b = \frac{1}{\sqrt{1156}}} \\{c = \frac{1}{\sqrt{1168}}}\end{matrix}.} \right.}} & (14)\end{matrix}$

And then, where the transform E_(16×16) is given as equation (8), thescaling matrix K_(16×16) is represented as

$\begin{matrix}{K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & {\; c^{2}} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {ac} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {b\; c} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2}\end{matrix}\mspace{14mu} \right\rbrack}} & (15)\end{matrix}$

Where the transform E_(16×16) is given as equation (11), and the scalingmatrix K_(16×16) is represented as

$\begin{matrix}{K_{16 \times 16}\mspace{11mu} = \mspace{14mu} {{\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & {\; c^{2}} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{matrix}\mspace{11mu} \right\rbrack}.}} & (16)\end{matrix}$

From the above, the 2D integer transform of X into Y is discussed. Itshould be appreciated that an inverse 2D integer transform of Y into Xmay be easily implemented based on the above mentioned 2D transform of Xinto Y, and be represented as

$\begin{matrix}{X = {{E_{16 \times 16}^{T} \cdot Y \cdot E_{16 \times 16}} = {E_{16 \times 16}^{T} \cdot \begin{bmatrix}y_{0,0} & y_{0,1} & \cdots & y_{0,15} \\y_{1,0} & y_{1,1} & \; & y_{1,15} \\\vdots & \; & ⋰ & \vdots \\y_{15,0} & y_{15,1} & \cdots & y_{15,15}\end{bmatrix} \cdot {E_{16 \times 16}.}}}} & (17)\end{matrix}$

Since the inverse 2D integer transform of Y into X can be computed by asimilar way as stated above, the detailed description thereof is thenomitted.

FIG. 4 illustrates exemplarily an embodiment of an apparatus 1000 forthe implementation of the process as shown in FIG. 1. As is showedtherein, the apparatus 1000 comprises an assignment unit 10, atransforming unit 20, a quantization unit 30 and a scaling unit 40.

The assignment unit 10 is configured to retrieve a data matrix X_(16×16)to be transformed, and then to assign of the retrieved data to anintermediate matrix C_(16×16) under the rule set forth in equation (4)or equation (9).

The transform unit 20 carries out two order-8 transforms for the matrixC_(16×16) by using the conventional E_(8×8), which is showed in equation(6), so as to output the matrix Z_(16×16). E_(8×8) may be embedded inthe transform unit 20. Alternatively, E_(8×8) may be stored in any otherseparately memory (not shown). In this case, the transform unit 20 maybe capable of retrieving E_(8×8) from the memory.

The quantization unit 30 receives the output matrix Z_(16×16), and thenconverts the matrix Z_(16×16) into a quantized transform coefficientY_(Q)(i,j), which requires less bits for representation.

The scaling unit 40 receives the quantized coefficient Y_(Q)(ij), andthen uses a scaling matrix K_(16×16) to make Y_(Q)(i,j) into a matrixY_(16×16), as is shown in equation (13), wherein matrix K_(16×16) isshown in equation (15) or equation (16). Herein, K_(16×16) may beembedded in the quantization unit 30. Alternatively, K_(16×16) may bestored in any other separately memory (not shown). In this case, thequantization unit 30 may be capable of retrieving K_(16×16) from thememory.

While the assignment unit 10, the transforming unit 20, the quantizationunit 30 and the scaling unit 40 are shown in FIG. 4 and described hereinas four separate units, other implementations are possible. For example,the units 10, 20, 30 and 40 may be implemented as lesser or more unitsas required, and may be implemented either by software or hardware, orthe combination of software and hardware.

The disclosed and other embodiments and the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, or in computer software, firmware, or hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The disclosedand other embodiments can be implemented as one or more computer programproducts, i.e., one or more modules of computer program instructionsencoded on a computer readable medium for execution by, or to controlthe operation of, data processing apparatus. The computer readablemedium can be a machine-readable storage device, a machine-readablestorage substrate, a memory device, a composition of matter effecting amachine-readable propagated signal, or a combination of one or morethem. The term “data processing apparatus” encompasses all apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, or multiple processors or computers.The apparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them. A propagated signal is an artificially generated signal, e.g.,a machine-generated electrical, optical, or electromagnetic signal, thatis generated to encode information for transmission to suitable receiverapparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of non volatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto optical disks; and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

The disclosed embodiments can be implemented in a computing system thatincludes a back end component, e.g., as a data server, or that includesa middleware component, e.g., an application server, or that includes afront end component, e.g., a client computer having a graphical userinterface or a Web browser through which a user can interact with animplementation of what is disclosed here, or any combination of one ormore such back end, middleware, or front end components. The componentsof the system can be interconnected by any form or medium of digitaldata communication, e.g., a communication network. Examples ofcommunication networks include a local area network (“LAN”) and a widearea network (“WAN”), e.g., the Internet.

A computer system for implementing the disclosed embodiments can includeclient computers (clients) and server computers (servers). A client anda server are generally remote from each other and typically interactthrough a communication network. The relationship of client and servercan arise by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other.

While this specification contains many specifics, these should not beconstrued as limitations on the scope of any invention or of what may beclaimed, but rather as descriptions of features specific to particularembodiments. Certain features that are described in this specificationin the context of separate embodiments can also be implemented incombination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, operations are depicted in the drawings in a particularorder, and such operations should be performed in the particular ordershown or in sequential order, and that all illustrated operations beperformed, to achieve desirable results. In certain circumstances,multitasking and parallel processing may be advantageous. Moreover, theseparation of various system components in the embodiments describedabove should not be understood as requiring such separation in allembodiments, and it should be understood that the described programcomponents and systems can generally be integrated together in a singlesoftware product or packaged into multiple software products.

Thus, particular embodiments have been described. Other embodiments arewithin the scope of the following claims.

1. A process for deriving an order-16 integer transform from an order-8integer transform in image and video coding, comprising: retrieving adata matrix X_(16×16) to be transformed; assigning the retrieved matrixX_(16×16) to an intermediate matrix C_(16×16) by eight additions andeight subtractions; and deriving a resultant matrix Z_(16×16) bycarrying out the order-8 integer transform twice by a rule of[z _(0,m) z _(1,m) . . . z _(7,m)]^(T) =E _(8×8) [c _(o,m) c _(1,m) . .. c _(7,m)]^(T)[z _(8,m) z _(9,m) . . . z _(15,m)]^(T) =E _(8×8) [c _(8,m) c _(9,m) . .. c _(15,m)]^(T) wherein z_(n,m) is of the (n,m)th element of the matrixZ_(16×16), and E_(8×8) is a standard order-8 transform used in thestandard of Microsoft Media Video.
 2. The process according to claim 1,wherein the assigning is carried out according to a rule of$C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\cdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\\cdots \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\cdots \\{x_{14,m} + x_{15,m}} \\{x_{0,m} - x_{1,m}} \\{x_{2,m} - x_{3,m}} \\\cdots \\{x_{14,m} - x_{15,m}}\end{bmatrix}}$ wherein C_(m) is the mth vector of the matrix C_(16×16),c_(i,m) is the (i, m)th element of C_(16×16), and x_(i,m) is the (i,m)th element of X_(16×16), 0≦m≦15 and 0≦i≦15.
 3. The process accordingto claim 1, wherein the assigning is carried out according to a rule of$C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\vdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\c_{10,m} \\c_{11,m} \\c_{12,m} \\c_{13,m} \\c_{14,m} \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\vdots \\{x_{14,m} + x_{15,m}} \\{x_{0,m} + x_{1,m}} \\{x_{3,m} + x_{2,m}} \\{x_{4,m} + x_{5,m}} \\{x_{7,m} + x_{6,m}} \\{x_{8,m} + x_{9,m}} \\{x_{11,m} + x_{10,m}} \\{x_{12,m} + x_{13,m}} \\{x_{15,m} + x_{14,m}}\end{bmatrix}}$ wherein C_(m) is the mth vector of the matrix C_(16×16),c_(i,m) is the (i, m)th element of C_(16×16), and x_(i,m) is the (i,m)th element of X_(16×16), 0≦m≦15 and 0≦i≦15.
 4. The process accordingto claim 2, further comprising: quantizing the resultant matrixZ_(16×16) by using a quantization parameter used in the Microsoft MediaVideo.
 5. The process according to claim 3, further comprising:quantizing the resultant matrix Z_(16×16) by using a quantizationparameter used in the Microsoft Media Video.
 6. The process according toclaim 4, further comprising: normalizing the quantized matrix by using ascaling matrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$7. The process according to claim 5, further comprising: normalizing thequantized matrix by using a scaling matrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$8. A method for processing an order-16 transform in image and videocoding, comprising: receiving a data matrix X_(16×16) to be transformed;and transforming the received data matrix X_(16×16) with an order-16transform matrix E_(16×16) to generate an output matrix Z_(16×16) by arule of Z_(16×16)=E_(16×16)X_(16×16), wherein the transform matrixE_(16×16) is represented as $E_{16 \times 16} = {\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 16 & 15 & 15 & 9 & 9 & 4 & 4 & {- 4} & {- 4} & {- 9} & {- 9} & {- 15} & {- 15} & {- 16} & {- 16} \\16 & 16 & 6 & 6 & {- 6} & {- 6} & {- 16} & {- 16} & {- 16} & {- 16} & {- 6} & {- 6} & 6 & 6 & 16 & 16 \\15 & 15 & {- 4} & {- 4} & {- 16} & {- 16} & {- 9} & {- 9} & 9 & 9 & 16 & 16 & 4 & 4 & {- 15} & {- 15} \\12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 & 12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 \\9 & 9 & {- 16} & {- 16} & 4 & 4 & 15 & 15 & {- 15} & {- 15} & {- 4} & {- 4} & 16 & {- 16} & {- 9} & {- 9} \\6 & 6 & {- 16} & {- 16} & 16 & 16 & {- 6} & {- 6} & {- 6} & {- 6} & 16 & 16 & {- 16} & {- 16} & 6 & 6 \\4 & 4 & {- 9} & {- 9} & 15 & 15 & {- 16} & {- 16} & 16 & 16 & {- 15} & {- 15} & 9 & 9 & {- 4} & {- 4} \\4 & {- 4} & {- 9} & 9 & 15 & {- 15} & {- 16} & 16 & 16 & {- 16} & {- 15} & 15 & 9 & {- 9} & {- 4} & 4 \\6 & {- 6} & {- 16} & 16 & 16 & {- 16} & {- 6} & 6 & {- 6} & 6 & 16 & {- 16} & {- 16} & 16 & 6 & {- 6} \\9 & {- 9} & {- 16} & 16 & 4 & {- 4} & 15 & {- 15} & {- 15} & 15 & {- 4} & 4 & 16 & {- 16} & {- 9} & 9 \\12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} & 12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} \\15 & {- 15} & {- 4} & 4 & {- 16} & 16 & {- 9} & 9 & 9 & {- 9} & 16 & {- 16} & 4 & {- 4} & {- 15} & 15 \\16 & {- 16} & 6 & {- 6} & {- 6} & 6 & {- 16} & 16 & {- 16} & 16 & {- 6} & 6 & 6 & {- 6} & 16 & {- 16} \\16 & {- 16} & 15 & {- 15} & 9 & {- 9} & 4 & {- 4} & {- 4} & 4 & {- 9} & 9 & {- 15} & 15 & {- 16} & 16 \\12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12}\end{bmatrix}.}$
 9. The method according to claim 8, further comprising:quantizing the output matrix Z_(16×16) by using a quantization parameterused in the Microsoft Media Video.
 10. The process according to claim 9,further comprising: normalizing the quantized matrix by using a scalingmatrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{14mu} {\frac{1}{2}\mspace{14mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$11. A method for processing an order-16 transform in image and videocoding, comprising: receiving a data matrix X_(16×16) to be transformed;and transforming the received data matrix X_(16×16) with an order-16transform matrix E_(16×16) to generate an output matrix Z_(16×16) by arule of Z_(16×16)=E_(16×16)X_(16×16), wherein the transform matrixE_(16×16) is represented as$E_{16 \times 16} = {{\frac{1}{2}\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 16 & 15 & 15 & 9 & 9 & 4 & 4 & {- 4} & {- 4} & {- 9} & {- 9} & {- 15} & {- 15} & {- 16} & {- 16} \\16 & 16 & 6 & 6 & {- 6} & {- 6} & {- 16} & {- 16} & {- 16} & {- 16} & {- 6} & {- 6} & 6 & 6 & 16 & 16 \\15 & 15 & {- 4} & {- 4} & {- 16} & {- 16} & {- 9} & {- 9} & 9 & 9 & 16 & 16 & 4 & 4 & {- 15} & {- 15} \\12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 & 12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 \\9 & 9 & {- 16} & {- 16} & 4 & 4 & 15 & 15 & {- 15} & {- 15} & {- 4} & {- 4} & 16 & 16 & {- 9} & {- 9} \\6 & 6 & {- 16} & {- 16} & 16 & 16 & {- 6} & {- 6} & {- 6} & {- 6} & 16 & 16 & {- 16} & {- 16} & 6 & 6 \\4 & 4 & {- 9} & {- 9} & 15 & 15 & {- 16} & {- 16} & 16 & 16 & {- 15} & {- 15} & 9 & 9 & {- 4} & {- 4} \\12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 & 12 & {- 12} & {- 12} & 12 \\16 & {- 16} & {- 15} & 15 & 9 & {- 9} & {- 4} & 4 & {- 4} & 4 & 9 & {- 9} & {- 15} & 15 & 16 & {- 16} \\16 & {- 16} & {- 6} & 6 & 6 & {- 6} & 16 & {- 16} & {- 16} & 16 & {- 6} & 6 & 6 & {- 6} & {- 16} & 16 \\15 & {- 15} & {- 4} & 4 & {- 16} & 16 & 9 & {- 9} & 9 & {- 9} & {- 16} & 16 & {- 4} & 4 & 15 & {- 15} \\12 & {- 12} & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & {- 12} & 12 \\9 & {- 9} & 16 & {- 16} & {- 4} & 4 & {- 15} & 15 & {- 15} & 15 & {- 4} & 4 & 16 & {- 16} & 6 & {- 6} \\6 & {- 6} & 16 & {- 16} & 16 & {- 16} & 6 & {- 6} & {- 6} & 6 & {- 16} & 16 & {- 16} & 16 & {- 6} & 6 \\4 & {- 4} & 9 & {- 9} & 15 & {- 15} & 16 & {- 16} & 16 & {- 16} & 15 & {- 15} & 9 & {- 9} & 4 & {- 4}\end{bmatrix}}.}$
 12. The method according to claim 11, furthercomprising: quantizing the output matrix by using a quantizationparameter used in the Microsoft Media Video.
 13. The process accordingto claim 12, further comprising: normalizing the quantized matrix byusing a scaling matrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{14mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$14. An apparatus for deriving an order-16 integer transform from anorder-8 integer transform in image and video coding, comprising: anassignment unit configured to retrieve a data matrix X_(16×16) to betransformed, and then to assign the retrieved matrix X_(16×16) to anintermediate matrix C_(16×16) by eight additions and eight subtractions;and a transform unit configured to derive a resultant matrix Z_(16×16)by carrying out the order-8 integer transform twice by a rule of[z _(0,m) z _(1,m) . . . z _(7,m)]^(T) =E _(8×8) [c _(0,m) c _(1,m) . .. c _(7,m)]^(T)[z _(8,m) z _(9,m) . . . z _(15,m)]^(T) =E _(8×8) [c _(8,m) c _(9,m) . .. c _(15,m)]^(T) wherein z_(n,m) is of the (n,m)th element of the matrixZ_(16×16), and E_(8'38) is a standard order-8 transform used in thestandard of Microsoft Media Video.
 15. The apparatus according to claim14, wherein the assignment unit is configured to assign the retrievedmatrix X_(16×16) to an intermediate matrix C_(16×16) according to a ruleof $C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\cdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\\cdots \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\cdots \\{x_{14,m} + x_{15,m}} \\{x_{0,m} - x_{1,m}} \\{x_{2,m} - x_{3,m}} \\\cdots \\{x_{14,m} - x_{15,m}}\end{bmatrix}}$ wherein C_(m) is the mth vector of the matrix C_(16×16),c_(i,m) is the (i, m)th element of C_(16×16), and x_(i,m) is the (i,m)th element of X_(16×16), 0≦m≦15 and 0≦i≦15.
 16. The apparatusaccording to claim 14, wherein the assignment unit is configured toassign the retrieved matrix X_(16×16) to an intermediate matrixC_(16×16) according to a rule of $C_{m} = {\begin{bmatrix}c_{0,m} \\c_{1,m} \\\vdots \\c_{7,m} \\c_{8,m} \\c_{9,m} \\c_{10,m} \\c_{11,m} \\c_{12,m} \\c_{13,m} \\c_{14,m} \\c_{15,m}\end{bmatrix} = \begin{bmatrix}{x_{0,m} + x_{1,m}} \\{x_{2,m} + x_{3,m}} \\\vdots \\{x_{14,m} - x_{15,m}} \\{x_{0,m} - x_{1,m}} \\{x_{3,m} - x_{2,m}} \\{x_{4,m} - x_{5,m}} \\{x_{7,m} - x_{6,m}} \\{x_{8,m} - x_{9,m}} \\{x_{11,m} - x_{10,m}} \\{x_{12,m} - x_{13,m}} \\{x_{15,m} - x_{14,m}}\end{bmatrix}}$ wherein C_(m) is the mth vector of the matrix C_(16×16),c_(i,m) is the (i, m)th element of C_(16×16), and c_(i,m) is the (i,m)th element of X_(16×16), 0≦m≦15 and 0≦i≦15.
 17. The apparatusaccording to claim 15, further comprising: a quantization unitconfigured to quantize the resultant matrix Z_(16×16) by using aquantization parameter used in the Microsoft Media Video.
 18. Theapparatus according to claim 16, further comprising: a quantization unitconfigured to quantize the resultant matrix Z_(16×16) by using aquantization parameter used in the Microsoft Media Video.
 19. Theapparatus according to claim 17, further comprising: a scaling unitconfigured to normalize the quantized matrix by using a scaling matrixK_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$20. The apparatus according to claim 18, further comprising: a scalingunit configured to normalize the quantized matrix by using a scalingmatrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$21. An apparatus for processing an order-16 transform in image and videocoding, comprising: a receiving unit configured to receive a data matrixX_(16×16) to be transformed; and a transform unit configured totransform the received data matrix X_(16×16) with an order-16 transformmatrix E_(16×16) to generate an output matrix Z_(16×16) by a rule ofZ_(16×16)=E_(16×16)X_(16×16), wherein the transform matrix E_(16×16) isrepresented as $E_{16 \times 16} = {\begin{bmatrix}12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\16 & 16 & 15 & 15 & 9 & 9 & 4 & 4 & {- 4} & {- 4} & {- 9} & {- 9} & {- 15} & {- 15} & {- 16} & {- 16} \\16 & 16 & 6 & 6 & {- 6} & {- 6} & {- 16} & {- 16} & {- 16} & {- 16} & {- 6} & {- 6} & 6 & 6 & 16 & 16 \\15 & 15 & {- 4} & {- 4} & {- 16} & {- 16} & {- 9} & {- 9} & 9 & 9 & 16 & 16 & 4 & 4 & {- 15} & {- 15} \\12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 & 12 & 12 & {- 12} & {- 12} & {- 12} & {- 12} & 12 & 12 \\9 & 9 & {- 16} & {- 16} & 4 & 4 & 15 & 15 & {- 15} & {- 15} & {- 4} & {- 4} & 16 & 16 & {- 9} & {- 9} \\6 & 6 & {- 16} & {- 16} & 16 & 16 & {- 6} & {- 6} & {- 6} & {- 6} & 16 & 16 & {- 16} & {- 16} & 6 & 6 \\4 & 4 & {- 9} & {- 9} & 15 & 15 & {- 16} & {- 16} & 16 & 16 & {- 15} & {- 15} & 9 & 9 & {- 4} & {- 4} \\4 & {- 4} & {- 9} & 9 & 15 & {- 15} & {- 16} & 16 & 16 & {- 16} & {- 15} & 15 & 9 & {- 9} & {- 4} & 4 \\6 & {- 6} & {- 16} & 16 & 16 & {- 16} & {- 6} & 6 & {- 6} & 6 & 16 & {- 16} & {- 16} & 16 & 6 & {- 6} \\9 & {- 9} & {- 16} & 16 & 4 & {- 4} & 15 & {- 15} & {- 15} & 15 & {- 4} & 4 & 16 & {- 16} & {- 9} & 9 \\12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} & 12 & {- 12} & {- 12} & 12 & {- 12} & 12 & 12 & {- 12} \\15 & {- 15} & {- 4} & 4 & {- 16} & 16 & {- 9} & 9 & 9 & {- 9} & 16 & {- 16} & 4 & {- 4} & {- 15} & 15 \\16 & {- 16} & 6 & {- 6} & {- 6} & 6 & {- 16} & 16 & {- 16} & 16 & {- 6} & 6 & 6 & {- 6} & 16 & {- 16} \\16 & {- 16} & 15 & {- 15} & 9 & {- 9} & 4 & {- 4} & {- 4} & 4 & {- 9} & 9 & {- 15} & 15 & {- 16} & 16 \\12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12} & 12 & {- 12}\end{bmatrix}.}$
 22. The apparatus according to claim 21, furthercomprising a quantizing unit configured to quantize the output matrixZ_(16×16) by using a quantization parameter used in the Microsoft MediaVideo.
 23. The apparatus according to claim 22, further comprising anormalizing unit configured to normalize the quantized matrix by using ascaling matrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{11mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$24. An apparatus for processing an order-16 transform in image and videocoding, comprising: a receiving unit configured to receive a data matrixX_(16×16) to be transformed; and a transform unit configured totransform the received data matrix X_(16×16) with an order-16 transformmatrix E_(16×16) to generate an output matrix Z_(16×16) by a rule ofZ_(16×16)=_(E) _(16×16)X_(16×16), wherein the transform matrix E_(16×16)is represented as $E_{16 \times 16} = {\begin{bmatrix}8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\10 & 10 & 9 & 9 & 6 & 6 & 2 & 2 & {- 2} & {- 2} & {- 6} & {- 6} & {- 9} & {- 9} & {- 10} & {- 10} \\10 & 10 & 4 & 4 & {- 4} & {- 4} & {- 10} & {- 10} & {- 10} & {- 10} & {- 4} & {- 4} & 4 & 4 & 10 & 10 \\9 & 9 & {- 2} & {- 2} & {- 10} & {- 10} & {- 6} & {- 6} & 6 & 6 & 10 & 10 & 2 & 2 & {- 9} & {- 9} \\8 & 8 & {- 8} & {- 8} & {- 8} & {- 8} & 8 & 8 & 8 & 8 & {- 8} & {- 8} & {- 8} & {- 8} & 8 & 8 \\6 & 6 & {- 10} & {- 10} & 2 & 2 & 9 & 9 & {- 9} & {- 9} & {- 2} & {- 2} & 10 & 10 & {- 6} & {- 6} \\4 & 4 & {- 10} & {- 10} & 10 & 10 & {- 4} & {- 4} & {- 4} & {- 4} & 10 & 10 & {- 10} & {- 10} & 4 & 4 \\2 & 2 & {- 6} & {- 6} & 9 & 9 & {- 10} & {- 10} & 10 & 10 & {- 9} & {- 9} & 6 & 6 & {- 2} & {- 2} \\8 & {- 8} & {- 8} & 8 & 8 & {- 8} & {- 8} & 8 & 8 & {- 8} & {- 8} & 8 & 8 & {- 8} & {- 8} & 8 \\10 & {- 10} & {- 9} & 9 & 6 & {- 6} & {- 2} & 2 & {- 2} & 2 & 6 & {- 6} & {- 9} & 9 & 10 & {- 10} \\10 & {- 10} & {- 4} & 4 & {- 4} & 4 & 10 & {- 10} & {- 10} & 10 & 4 & {- 4} & 4 & {- 4} & {- 10} & 10 \\9 & {- 9} & 2 & {- 2} & {- 10} & 10 & 6 & {- 6} & 6 & {- 6} & {- 10} & 10 & 2 & {- 2} & 9 & {- 9} \\8 & {- 8} & 8 & {- 8} & {- 8} & 8 & {- 8} & 8 & 8 & {- 8} & 8 & {- 8} & {- 8} & 8 & {- 8} & 8 \\6 & {- 6} & 10 & {- 10} & 2 & {- 2} & {- 9} & 9 & {- 9} & 9 & 2 & {- 2} & 10 & {- 10} & 6 & {- 6} \\4 & {- 4} & 10 & {- 10} & 10 & {- 10} & 4 & {- 4} & {- 4} & 4 & {- 10} & 10 & {- 10} & 10 & {- 4} & 4 \\2 & {- 2} & 6 & {- 6} & 9 & {- 9} & 10 & {- 10} & 10 & {- 10} & 9 & {- 9} & 6 & {- 6} & 2 & {- 2}\end{bmatrix}.}$
 25. The apparatus according to claim 24, furthercomprising a quantizing unit configured to quantize the output matrixZ_(16×16) by using a quantization parameter used in the Microsoft MediaVideo.
 26. The apparatus according to claim 25, further comprising anormalizing unit configured to normalize the quantized matrix by using ascaling matrix K_(16×16), where,$K_{16 \times 16}\mspace{11mu} = \mspace{11mu} {\frac{1}{2}\mspace{14mu}\left\lbrack \mspace{14mu} \begin{matrix}a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} & a^{2} & {ab} & {a\; c} & {ab} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} \\{a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} & {a\; c} & {bc} & c^{2} & {bc} \\{ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2} & {ab} & b^{2} & {bc} & b^{2}\end{matrix}\mspace{14mu} \right\rbrack}$${{{and}\mspace{14mu} {where}\mspace{14mu} a} = \frac{1}{\sqrt{1512}}},{b = {{\frac{1}{\sqrt{1156}}\mspace{14mu} {and}\mspace{14mu} c} = {\frac{1}{\sqrt{1168}}.}}}$